Question

8. On vacation, you fish for walleye on Lake Lena, i northern Minnesota, where the length in inches of caught walleye is normally distributed with mean 18 and standard deviation 4. (c) Minnesota law says that you must release all walleye between 12 and 22 inches long. Find the probability that a given fish you catch is legal to keep. (d) You quit for the day as soon as you've caught 5 legal fish. What's the expected number of fish you'll need to catch? (Explain with reference to an appropriate RV.)

Answer 1

Let X be the RV denoting the length. Then X ~N(18,42). (c) Now probability that a given fish is between 12 and 22 inches is (12-N <x-18 <2-18) P(12 < X < 22) = P = P(-1.5 < Z < 1) = Φ(1)-Φ(-1.5) = 0.8413-0.0668 = 0.7745. In the above calculation we use that Z- distribution and we get the values of Φ(1) and φ(-1.5) from the table Hence the probability that a given fish you catch is legal to keep is 1 - 0.77450.2255 X18 follows standard normal (d) Consider a random variable Y which denotes the expected number of fish caught until a legal fish is caught. Then y follows a Geometric(p) distribution, where p 0.2255. Then E Y]235 Now let Yi be the number of fish caught until the first legal fish caught, Y2 be the number of fish caught after that until the second legal fish is caught and so on. We want ΕΙΣ5-1 , Yi. Bv linearity of expectation we have 0.22550.2255